Return to Question

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\rVert \end{align} This minimization problem can be solved for $K=\mathbb{R},K=\mathbb{C}$ with the SVD-algorithm. The corrosponding right-singular vector of the smallest singular value is a solution (If $A=U\Sigma V^*$ is the SVD and the singular values on the diagonal of $\Sigma$ are sorted by their absolute value then the rightmost column of $V$ solves the problem).

However, I am now looking for a more general method of solving this problem for any field $K$. Is there a generalization of the SVD-algorithm that maybe preserves this property?

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\rVert \end{align} This minimization problem can be solved for $K=\mathbb{R},K=\mathbb{C}$ with the SVD-algorithm. The corrosponding right-singular vector of the smallest singular value is a solution (If $A=U\Sigma V^*$ is the SVD and the singular values on the diagonal of $\Sigma$ are sorted by their absolute value then the rightmost column of $V$ solves the problem).

However, I am now looking for a more general method of solving this problem for any field $K$. Is there a generalization of the SVD-algorithm that maybe preserves this property?

Edit: The same method works for algebraic numbers (The singular value problem can be reduced to the eigenvalue problem which is closed under the algebraic numbers). The comments pointed quaternionics as well.

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ a norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\rVert \end{align} This minimization problem can be solved for $K=\mathbb{R},K=\mathbb{C}$ with the SVD-algorithm. The corrosponding right-singular vector of the smallest singular value is a solution (If $A=U\Sigma V^*$ is the SVD and the singular values on the diagonal of $\Sigma$ are sorted by their absolute value then the rightmost column of $V$ solves the problem).

However, I am now looking for a more general method of solving this problem for any field $K$. Is there a generalization of the SVD-algorithm that maybe preserves this property?

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\rVert \end{align} This minimization problem can be solved for $K=\mathbb{R},K=\mathbb{C}$ with the SVD-algorithm. The corrosponding right-singular vector of the smallest singular value is a solution (If $A=U\Sigma V^*$ is the SVD and the singular values on the diagonal of $\Sigma$ are sorted by their absolute value then the rightmost column of $V$ solves the problem).

However, I am now looking for a more general method of solving this problem for any field $K$. Is there a generalization of the SVD-algorithm that maybe preserves this property?

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ a norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\rVert \end{align} This minimization problem can be solved for $K=\mathbb{R}$ with the SVD-algorithm. The corrosponding right-singular vector of the smallest singular value is a solution (If $A=U\Sigma V^*$ is the SVD and the singular values on the diagonal of $\Sigma$ are sorted by their absolute value then the rightmost column of $V$ solves the problem).

However, I am now looking for a more general method of solving this problem for any field $K$. Is there a generalization of the SVD-algorithm that maybe preserves this property?

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ a norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\rVert \end{align} This minimization problem can be solved for $K=\mathbb{R},K=\mathbb{C}$ with the SVD-algorithm. The corrosponding right-singular vector of the smallest singular value is a solution (If $A=U\Sigma V^*$ is the SVD and the singular values on the diagonal of $\Sigma$ are sorted by their absolute value then the rightmost column of $V$ solves the problem).

However, I am now looking for a more general method of solving this problem for any field $K$. Is there a generalization of the SVD-algorithm that maybe preserves this property?